CGAL 4.9 - Manual
Package Overview

# Arithmetic and Algebra

## Algebraic Foundations

Michael Hemmer
This package defines what algebra means for CGAL, in terms of concepts, classes and functions. The main features are: (i) explicit concepts for interoperability of types (ii) separation between algebraic types (not necessarily embeddable into the reals), and number types (embeddable into the reals).
Introduced in: CGAL 3.3
BibTeX: cgal:h-af-16b

## Number Types

Michael Hemmer, Susan Hert, Sylvain Pion, and Stefan Schirra
This package provides number type concepts as well as number type classes and wrapper classes for third party number type libraries.
Introduced in: CGAL 1.0
BibTeX: cgal:hhkps-nt-16b

## Modular Arithmetic

Michael Hemmer and Sylvain Pion
This package provides arithmetic over finite fields. The provided tools are in particular useful for filters based on modular arithmetic and algorithms based on Chinese remainder.
Introduced in: CGAL 3.4
BibTeX: cgal:h-ma-16b

## Polynomial

Michael Hemmer
This package introduces a concept for univariate and multivariate polynomials in $$d$$ variables. Though the concept is written for an arbitrary number of variables, the number of variables is considered as fixed for a particular model of this concept.
Introduced in: CGAL 3.3
BibTeX: cgal:h-p-16b

## Algebraic Kernel

Eric Berberich, Michael Hemmer, Michael Kerber, Sylvain Lazard, Luis Peñaranda, and Monique Teillaud
Real solving of polynomials is a fundamental problem with a wide application range. This package is targeted to provide black-box implementations of state-of-the-art algorithms to determine, compare and approximate real roots of univariate polynomials and bivariate polynomial systems. Such a black-box is called an Algebraic Kernel. So far the package only provides models for the univariate kernel. Nevertheless, it already defines concepts for the bivariate kernel, since this settles the interface for upcoming implementations.
Introduced in: CGAL 3.6
Depends on: Some models depend on RS and RS3.
BibTeX: cgal:bht-ak-16b

# Combinatorial Algorithms

## Monotone and Sorted Matrix Search

Michael Hoffmann
This package provides a matrix search framework, which is the underlying technique for the computation of all furthest neighbors for the vertices of a convex polygon, maximal k-gons inscribed into a planar point set, and computing rectangular p-centers.
Introduced in: CGAL 1.1
BibTeX: cgal:h-msms-16b

## Linear and Quadratic Programming Solver

Kaspar Fischer, Bernd Gärtner, Sven Schönherr, and Frans Wessendorp
This package contains algorithms for minimizing linear and convex quadratic functions over polyhedral domains, described by linear equations and inequalities. The algorithms are exact, i.e. the solution is computed in terms of multiprecision rational numbers. The resulting solution is certified: along with the claims that the problem under consideration has an optimal solution, is infeasible, or is unbounded, the algorithms also deliver proofs for these facts. These proofs can easily (and independently from the algorithms) be checked for correctness. The solution algorithms are based on a generalization of the simplex method to quadratic objective functions.
Introduced in: CGAL 3.3
BibTeX: cgal:fgsw-lqps-16b

# Geometry Kernels

## 2D and 3D Linear Geometry Kernel

Hervé Brönnimann, Andreas Fabri, Geert-Jan Giezeman, Susan Hert, Michael Hoffmann, Lutz Kettner, Sylvain Pion, and Stefan Schirra
This package contains kernels each containing objects of constant size, such as point, vector, direction, line, ray, segment, circle as well as predicates and constructions for these objects. The kernels mainly differ in the way they handle robustness issues.
Introduced in: CGAL 0.9
BibTeX: cgal:bfghhkps-lgk23-16b

## dD Geometry Kernel

Michael Seel
The dD Kernel contains objects of constant size, such as point, vector, direction, line, ray, segment, circle in d dimensional Euclidean space, as well as predicates and constructions for these objects.
Introduced in: CGAL 1.1
BibTeX: cgal:s-gkd-16b

## 2D Circular Geometry Kernel

Pedro Machado Manhães de Castro, Sylvain Pion, and Monique Teillaud
This package is an extension of the linear CGAL kernel. It offers functionalities on circles, circular arcs and line segments in the plane.
Introduced in: CGAL 3.2
BibTeX: cgal:cpt-cgk2-16b
Windows Demo: Arrangement of Circular Arcs
Common Demo Dlls: dlls

## 3D Spherical Geometry Kernel

Pedro Machado Manhães de Castro, Frédéric Cazals, Sébastien Loriot, and Monique Teillaud
This package is an extension of the linear CGAL Kernel. It offers functionalities on spheres, circles, circular arcs and line segments, in the 3D space or restricted on a reference sphere.
Introduced in: CGAL 3.4
BibTeX: cgal:cclt-sgk3-16b

# Convex Hull Algorithms

## 2D Convex Hulls and Extreme Points

Susan Hert and Stefan Schirra
This package provides functions for computing convex hulls in two dimensions as well as functions for checking if sets of points are strongly convex are not. There are also a number of functions described for computing particular extreme points and subsequences of hull points, such as the lower and upper hull of a set of points.
Introduced in: CGAL 1.0
BibTeX: cgal:hs-chep2-16b
Windows Demo: See Bounding Volumes
Common Demo Dlls: dlls

## 3D Convex Hulls

Susan Hert and Stefan Schirra
This package provides functions for computing convex hulls in three dimensions as well as functions for checking if sets of points are strongly convex or not. One can compute the convex hull of a set of points in three dimensions in two ways: using a static algorithm or using a triangulation to get a fully dynamic computation.
Introduced in: CGAL 1.1
Depends on: All algorithms produce as output a Polyhedron. The dynamic algorithms depend on 3D Triangulations.
BibTeX: cgal:hs-ch3-16b
Windows Demo: See Polyhedral Surface
Common Demo Dlls: dlls

## dD Convex Hulls and Delaunay Triangulations

Susan Hert and Michael Seel
This package provides functions for computing convex hulls and Delaunay triangulations in $$d$$-dimensional Euclidean space.
Introduced in: CGAL 2.3
BibTeX: cgal:hs-chdt3-16b

# Polygons

## 2D Polygons

This package provides a 2D polygon class and operations on sequences of points, like bounding box, extremal points, signed area, simplicity and convexity test, orientation, and point location. The demo includes operations on polygons, such as computing a convex partition, and the straight skeleton.
Introduced in: CGAL 0.9
BibTeX: cgal:gw-p2-16b
Windows Demo: Operations on Polygons
Common Demo Dlls: dlls

## 2D Regularized Boolean Set-Operations

Efi Fogel, Ophir Setter, Ron Wein, Guy Zucker, Baruch Zukerman, and Dan Halperin
This package consists of the implementation of Boolean set-operations on point sets bounded by weakly x-monotone curves in 2-dimensional Euclidean space. In particular, it contains the implementation of regularized Boolean set-operations, intersection predicates, and point containment predicates.
Introduced in: CGAL 3.2
Depends on: 2D Arrangements
BibTeX: cgal:fwzh-rbso2-16b
Windows Demo: Boolean operations
Common Demo Dlls: dlls

## 2D Boolean Operations on Nef Polygons

Michael Seel
A Nef polygon is any set that can be obtained from a finite set of open halfspaces by set complement and set intersection operations. Due to the fact that all other binary set operations like union, difference and symmetric difference can be reduced to intersection and complement calculations, Nef polygons are also closed under those operations. Apart from the set complement operation there are more topological unary set operations that are closed in the domain of Nef polygons interior, boundary, and closure.
Introduced in: CGAL 2.3
BibTeX: cgal:s-bonp2-16b
Windows Demo: 2D Nef Polygons
Common Demo Dlls: dlls

## 2D Boolean Operations on Nef Polygons Embedded on the Sphere

Peter Hachenberger and Lutz Kettner
This package offers the equivalent to 2D Nef Polygons in the plane. Here halfplanes correspond to half spheres delimited by great circles.
Introduced in: CGAL 3.1
Depends on: 2D Boolean Operations on Nef Polygons
BibTeX: cgal:hk-bonpes2-16b

## 2D Polygon Partitioning

Susan Hert
This package provides functions for partitioning polygons in monotone or convex polygons. The algorithms can produce results with the minimal number of polygons, as well as approximations which have no more than four times the optimal number of convex pieces but they differ in their runtime complexities.
Introduced in: CGAL 2.3
BibTeX: cgal:h-pp2-16b
Windows Demos: 2D Polygon Partition, Operations on Polygons
Common Demo Dlls: dlls

## 2D Straight Skeleton and Polygon Offsetting

Fernando Cacciola
This package implements an algorithm to construct a halfedge data structure representing the straight skeleton in the interior of 2D polygons with holes and an algorithm to construct inward offset polygons at any offset distance given a straight skeleton.
Introduced in: CGAL 3.2
Depends on: Halfedge Data Structures
BibTeX: cgal:c-sspo2-16b
Windows Demos: 2D Straight Skeleton, Operations on Polygons
Common Demo Dlls: dlls

## 2D Minkowski Sums

Ron Wein, Alon Baram, Eyal Flato, Efi Fogel, Michael Hemmer, Sebastian Morr
This package consists of functions that compute the Minkowski sum of two simple straight-edge polygons in the plane. It also contains functions for computing the Minkowski sum of a polygon and a disc, an operation known as offsetting or dilating a polygon. The package can compute the exact representation of the offset polygon, or provide a guaranteed approximation of the offset.

## 2D Polyline Simplification

Andreas Fabri
This package enables to simplify polylines with the guarantee that the topology of the polylines does not change. This can be done for a single polyline as well as for a set of polyline constraints in a constrained triangulation. The simplification can be controlled with cost and stop functions.
Introduced in: CGAL 4.6
Depends on: 2D Triangulation
BibTeX: cgal:f-ps2-16b
Windows Demo: Polyline Simplification
Common Demo Dlls: dlls

## 2D Visibility Computation

Michael Hemmer, Kan Huang, Francisc Bungiu, Ning Xu
This package provides several variants to compute the visibility area of a point within polygonal regions in two dimensions.
Introduced in: CGAL 4.7
Depends on: 2D Arrangements
Depends on: 2D Triangulation
BibTeX: hhb-visibility-2-16b

# Cell Complexes and Polyhedra

## 3D Polyhedral Surface

Lutz Kettner
Polyhedral surfaces in three dimensions are composed of vertices, edges, facets and an incidence relationship on them. The organization beneath is a halfedge data structure, which restricts the class of representable surfaces to orientable 2-manifolds - with and without boundary. If the surface is closed we call it a polyhedron.
Introduced in: CGAL 1.0
Depends on: Halfedge Data Structures
BibTeX: cgal:k-ps-16b
Windows Demo: Operations on Polyhedra
Common Demo Dlls: dlls

## Halfedge Data Structures

Lutz Kettner
A halfedge data structure is an edge-centered data structure capable of maintaining incidence information of vertices, edges and faces, for example for planar maps, polyhedra, or other orientable, two-dimensional surfaces embedded in arbitrary dimension. Each edge is decomposed into two halfedges with opposite orientations. One incident face and one incident vertex are stored in each halfedge. For each face and each vertex, one incident halfedge is stored. Reduced variants of the halfedge data structure can omit some of these information, for example the halfedge pointers in faces or the storage of faces at all.
Introduced in: CGAL 1.0
BibTeX: cgal:k-hds-16b

## Surface Mesh

Mario Botsch, Daniel Sieger, Philipp Moeller, and Andreas Fabri
The surface mesh class provided by this package is an implementation of the halfedge data structure allowing to represent polyhedral surfaces. It is an alternative to the packages Halfedge Data Structures and 3D Polyhedral Surface. The main differences are that it is indexed based and not pointer based, and that the mechanism for adding information to vertices, halfedges, edges, and faces is much simpler and can be used at runtime and not at compile time.
Introduced in: CGAL 4.6
BibTeX: cgal:bsmf-sm-16b

## Combinatorial Maps

Guillaume Damiand
This package implements Combinatorial Maps in d-dimension. A combinatorial map is a data structure allowing to represent an orientable subdivided object by describing all the cells of the subdivision (for example in 3D vertices, edges, faces, volumes) and all the incidence and adjacency relationships between these cells. Information can be associated to cells thanks to attributes. In 2D, a combinatorial map is equivalent to a halfedge data structure. The package provides basic creations, modification operations, and several iterators allowing to run through some specific part of the object.
Introduced in: CGAL 3.9
BibTeX: cgal:d-cm-16b

## Linear Cell Complex

Guillaume Damiand
This package implements linear cell complexes, objects in d-dimension with linear geometry. The combinatorial part of objects is described by a combinatorial map, representing all the cells of the object plus the incidence and adjacency relations between cells. Geometry is added to combinatorial maps simply by associating a point to each vertex of the map. Taking a 2D combinatorial map, and using 3D points, gives a linear cell complex equivalent to a Polyhedron_3.
Introduced in: CGAL 4.0
Depends on: Combinatorial Maps
BibTeX: cgal:d-lcc-12-16b
Windows Demo: 3D Linear Cell Complex
Common Demo Dlls: dlls

## 3D Boolean Operations on Nef Polyhedra

Peter Hachenberger and Lutz Kettner
3D Nef polyhedra, are a boundary representation for cell-complexes bounded by halfspaces that supports Boolean operations and topological operations in full generality including unbounded cells, mixed dimensional cells (e.g., isolated vertices and antennas). Nef polyhedra distinguish between open and closed sets and can represent non-manifold geometry.

## Convex Decomposition of Polyhedra

Peter Hachenberger
This packages provides a function for decomposing a bounded polyhedron into convex sub-polyhedra. The decomposition yields $$O(r^2)$$ convex pieces, where $$r$$ is the number of edges, whose adjacent facets form an angle of more than 180 degrees with respect to the polyhedron's interior. This bound is worst-case optimal.
Introduced in: CGAL 3.5
BibTeX: cgal:h-emspe-16b
Windows Demo: Operations on Polyhedra
Common Demo Dlls: dlls

## 3D Minkowski Sum of Polyhedra

Peter Hachenberger
This package provides a function, which computes the Minkowski sum of two point sets in $$\mathbb{R}^3$$. These point sets may consist of isolated vertices, isolated edges, surfaces with convex facets without holes, and open and closed solids. Thus, it is possible to compute the configuration space of translational robots (even in tight passage scenarios) as well as several graphics operations, like for instance the glide operation, which computes the point set swept by a polyhedron that moves along a polygonal line.
Introduced in: CGAL 3.5
Depends on: 3D Boolean Operations on Nef Polyhedra, Convex Decomposition of Polyhedra
BibTeX: cgal:h-msp3-16b
Windows Demo: Operations on Polyhedra
Common Demo Dlls: dlls

# Arrangements

## 2D Arrangements

Ron Wein, Eric Berberich, Efi Fogel, Dan Halperin, Michael Hemmer, Oren Salzman, and Baruch Zukerman
This package can be used to construct, maintain, alter, and display arrangements in the plane. Once an arrangement is constructed, the package can be used to obtain results of various queries on the arrangement, such as point location. The package also includes generic implementations of two algorithmic frameworks, that are, computing the zone of an arrangement, and line-sweeping the plane, the arrangements is embedded on. These frameworks are used in turn in the implementations of other operations on arrangements. Computing the overlay of two arrangements, for example, is based on the sweep-line framework. Arrangements and arrangement components can also be extended to store additional data. An important extension stores the construction history of the arrangement, such that it is possible to obtain the originating curve of an arrangement subcurve.
Introduced in: CGAL 2.1
BibTeX: cgal:wfzh-a2-16b
Windows Demo: 2D Arrangements
Common Demo Dlls: dlls

## 2D Intersection of Curves

Baruch Zukerman, Ron Wein, and Efi Fogel
This package provides three free functions implemented based on the sweep-line paradigm: given a collection of input curves, compute all intersection points, compute the set of subcurves that are pairwise interior-disjoint induced by them, and check whether there is at least one pair of curves among them that intersect in their interior.
Introduced in: CGAL 2.4
Depends on: 2D Arrangements
BibTeX: cgal:wfz-ic2-16b

## 2D Snap Rounding

Eli Packer
Snap Rounding is a well known method for converting arbitrary-precision arrangements of segments into a fixed-precision representation. In the study of robust geometric computing, it can be classified as a finite precision approximation technique. Iterated Snap Rounding is a modification of Snap Rounding in which each vertex is at least half-the-width-of-a-pixel away from any non-incident edge. This package supports both methods.
Introduced in: CGAL 3.1
Depends on: 2D Arrangements
BibTeX: cgal:p-sr2-16b
Windows Demo: 2D Snap Rounding
Common Demo Dlls: dlls

## 2D Envelopes

Ron Wein
This package consits of functions that computes the lower (or upper) envelope of a set of arbitrary curves in 2D. The output is represented as an envelope diagram, namely a subdivision of the $$x$$-axis into intervals, such that the identity of the curves that induce the envelope on each interval is unique.
Introduced in: CGAL 3.3
Depends on: 2D Arrangements
BibTeX: cgal:w-e2-16b

## 3D Envelopes

Dan Halperin, Michal Meyerovitch, Ron Wein, and Baruch Zukerman
This package consits of functions that compute the lower (or upper) envelope of a set of arbitrary surfaces in 3D. The output is represented as an 2D envelope diagram, namely a planar subdivision such that the identity of the surfaces that induce the envelope over each diagram cell is unique.
Introduced in: CGAL 3.3
Depends on: 2D Arrangements
BibTeX: cgal:mwz-e3-16b
Windows Demos: 3D Envelopes, L1 Voronoi Diagram
Common Demo Dlls: dlls

# Triangulations and Delaunay Triangulations

## 2D Triangulation

Mariette Yvinec
This package allows to build and handle various triangulations for point sets two dimensions. Any CGAL triangulation covers the convex hull of its vertices. Triangulations are built incrementally and can be modified by insertion or removal of vertices. They offer point location facilities. The package provides plain triangulation (whose faces depend on the insertion order of the vertices) and Delaunay triangulations. Regular triangulations are also provided for sets of weighted points. Delaunay and regular triangulations offer nearest neighbor queries and primitives to build the dual Voronoi and power diagrams. Finally, constrained and Delaunay constrained triangulations allows to force some constrained segments to appear as edges of the triangulation. Several versions of constrained and Delaunay constrained triangulations are provided: some of them handle intersections between input constraints segment while others do not.
Introduced in: CGAL 0.9
Depends on: 2D Triangulation Data Structure
BibTeX: cgal:y-t2-16b
Windows Demos: Delaunay Triangulation, Regular Triangulation, Constrained Delaunay Triangulation
Common Demo Dlls: dlls

## 2D Triangulation Data Structure

Sylvain Pion and Mariette Yvinec
This package provides a data structure to store a two-dimensional triangulation that has the topology of a two-dimensional sphere. The package acts as a container for the vertices and faces of the triangulation and provides basic combinatorial operation on the triangulation.
Introduced in: CGAL 2.2
BibTeX: cgal:py-tds2-16b

## 2D Periodic Triangulations

Nico Kruithof
This package allows to build and handle triangulations of point sets in the two dimensional flat torus. Triangulations are built incrementally and can be modified by insertion or removal of vertices. They offer point location facilities. The package provides Delaunay triangulations and offers nearest neighbor queries and primitives to build the dual Voronoi diagrams.
Introduced in: CGAL 4.3
Depends on: 2D Triangulation
BibTeX: cgal:k-pt2-13-16b
Windows Demo: Periodic Delaunay Triangulation
Common Demo Dlls: dlls

## 3D Triangulations

Clément Jamin, Sylvain Pion and Monique Teillaud
This package allows to build and handle triangulations for point sets in three dimensions. Any CGAL triangulation covers the convex hull of its vertices. Triangulations are build incrementally and can be modified by insertion, displacements or removal of vertices. They offer point location facilities. The package provides plain triangulation (whose faces depends on the insertion order of the vertices) and Delaunay triangulations. Regular triangulations are also provided for sets of weighted points. Delaunay and regular triangulations offer nearest neighbor queries and primitives to build the dual Voronoi and power diagrams. Optionally, the main Delaunay and regular triangulation algorithms (insert, remove) support multi-core shared-memory architectures to take advantage of available parallelism.
Introduced in: CGAL 2.1
Depends on: 3D Triangulation Data Structure
BibTeX: cgal:pt-t3-16b
Windows Demo: 3D Triangulations
Common Demo Dlls: dlls

## 3D Triangulation Data Structure

Clément Jamin, Sylvain Pion and Monique Teillaud
This package provides a data structure to store a three-dimensional triangulation that has the topology of a three-dimensional sphere. The package acts as a container for the vertices and cells of the triangulation and provides basic combinatorial operations on the triangulation.
Introduced in: CGAL 2.1
BibTeX: cgal:pt-tds3-16b

## 3D Periodic Triangulations

Manuel Caroli and Monique Teillaud
This package allows to build and handle triangulations of point sets in the three dimensional flat torus. Triangulations are built incrementally and can be modified by insertion or removal of vertices. They offer point location facilities. The package provides Delaunay triangulations and offers nearest neighbor queries and primitives to build the dual Voronoi diagrams.
Introduced in: CGAL 3.5
Depends on: 3D Triangulation Data Structure
BibTeX: cgal:ct-pt3-16b
Windows Demos: 3D Periodic Delaunay Triangulation, 3D Periodic Lloyd
Common Demo Dlls: dlls

## dD Triangulations

Samuel Hornus, Olivier Devillers and Clément Jamin
This package provides classes for manipulating triangulations (pure simplicial complexes) in Euclidean spaces whose dimension can be specified at compile-time or at run-time. Specifically, it provides a data structure to store the triangulations, and two classes to handle triangulations and Delaunay triangulations of point sets. Point location and point insertion are supported. The Delaunay triangulation also supports point removal.
Introduced in: CGAL 4.6
BibTeX: cgal:hdj-t-16b

## 2D Alpha Shapes

Tran Kai Frank Da
This package offers a data structure encoding the whole family of alpha-complexes related to a given 2D Delaunay or regular triangulation. In particular, the data structure allows to retrieve the alpha-complex for any alpha value, the whole spectrum of critical alpha values and a filtration on the triangulation faces (this filtration is based on the first alpha value for which each face is included on the alpha-complex).
Introduced in: CGAL 2.1
Depends on: 2D Triangulation
BibTeX: cgal:d-as2-16b
Windows Demo: 2D Alpha Shapes
Common Demo Dlls: dlls

## 3D Alpha Shapes

Tran Kai Frank Da, Sébastien Loriot, and Mariette Yvinec
This package offers a data structure encoding either one alpha-complex or the whole family of alpha-complexes related to a given 3D Delaunay or regular triangulation. In the latter case, the data structure allows to retrieve the alpha-complex for any alpha value, the whole spectrum of critical alpha values and a filtration on the triangulation faces (this filtration is based on the first alpha value for which each face is included on the alpha-complex).
Introduced in: CGAL 2.3
Depends on: 3D Triangulations
BibTeX: cgal:dy-as3-16b
Windows Demo: 3D Alpha Shapes
Common Demo Dlls: dlls

# Voronoi Diagrams

## 2D Segment Delaunay Graphs

Menelaos Karavelas
An algorithm for computing the dual of a Voronoi diagram of a set of segments under the Euclidean metric. It is a generalization of the standard Voronoi diagram for points. The algorithms provided are dynamic.
Introduced in: CGAL 3.1
Depends on: 2D Triangulation Data Structure
BibTeX: cgal:k-sdg2-16b
Windows Demo: 2D Segment Voronoi Diagram
Common Demo Dlls: dlls

## L Infinity Segment Delaunay Graphs

Panagiotis Cheilaris, Sandeep Kumar Dey, Evanthia Papadopoulou
Algorithms and geometric traits for computing the dual of the Voronoi diagram of a set of points and segments under the $$L_{\infty}$$ metric.
Introduced in: CGAL 4.7
Depends on: 2D Segment Delaunay Graphs
BibTeX: cgal:cdp-sdglinf2-16b

## 2D Apollonius Graphs (Delaunay Graphs of Disks)

Menelaos Karavelas and Mariette Yvinec
Algorithms for computing the Apollonius graph in two dimensions. The Apollonius graph is the dual of the Apollonius diagram, also known as the additively weighted Voronoi diagram. The latter can be thought of as the Voronoi diagram of a set of disks under the Euclidean metric, and it is a generalization of the standard Voronoi diagram for points. The algorithms provided are dynamic.
Introduced in: CGAL 3.0
Depends on: 2D Triangulation Data Structure
BibTeX: cgal:ky-ag2-16b
Windows Demo: 2D Apollonius Graph
Common Demo Dlls: dlls

Menelaos Karavelas
The 2D Voronoi diagram adaptor package provides an adaptor that adapts a 2-dimensional triangulated Delaunay graph to the corresponding Voronoi diagram, represented as a doubly connected edge list (DCEL) data structure. The adaptor has the ability to automatically eliminate, in a consistent manner, degenerate features of the Voronoi diagram, that are artifacts of the requirement that Delaunay graphs should be triangulated even in degenerate configurations. Depending on the type of operations that the underlying Delaunay graph supports, the adaptor allows for the incremental or dynamic construction of Voronoi diagrams and can support point location queries.
Introduced in: CGAL 3.2
BibTeX: cgal:k-vda2-16b
Windows Demos: 2D Point Voronoi Diagram , 2D Disk Voronoi Diagram, 2D Segment Voronoi Diagram
Common Demo Dlls: dlls

# Mesh Generation

## 2D Conforming Triangulations and Meshes

Laurent Rineau
This package implements a Delaunay refinement algorithm to construct conforming triangulations and 2D meshes. Conforming Delaunay triangulations are obtained from constrained Delaunay triangulations by refining constrained edges until they are Delaunay edges. Conforming Gabriel triangulations are obtained by further refining constrained edges until they become Gabriel edges. The package provides also a 2D mesh generator that refines triangles and constrained edges until user defined size and shape criteria on triangles are satisfied. The generated meshes can be optimized using the Lloyd algorithm, also provided in this package. The package can handle intersecting input constraints and set no restriction on the angle formed by two constraints sharing an endpoint.
Introduced in: CGAL 3.1
Depends on: 2D Triangulation
BibTeX: cgal:r-ctm2-16b
Windows Demo: 2D Mesh Generator
Common Demo Dlls: dlls

## 3D Surface Mesh Generation

Laurent Rineau and Mariette Yvinec
This package provides functions to generate surface meshes that interpolate smooth surfaces. The meshing algorithm is based on Delaunay refinement and provides some guarantees on the resulting mesh: the user is able to control the size and shape of the mesh elements and the accuracy of the surface approximation. There is no restriction on the topology and number of components of input surfaces. The surface mesh generator may also be used for non smooth surfaces but without guarantee. Currently, implementations are provided for implicit surfaces described as the zero level set of some function and surfaces described as a gray level set in a three-dimensional image.
Introduced in: CGAL 3.2
Depends on: 3D Triangulations
BibTeX: cgal:ry-smg-16b
Windows Demo: Surface Mesh Generator
Common Demo Dlls: dlls

## 3D Skin Surface Meshing

Nico Kruithof
This package allows to build a triangular mesh of a skin surface. Skin surfaces are used for modeling large molecules in biological computing. The surface is defined by a set of balls, representing the atoms of the molecule, and a shrink factor that determines the size of the smooth patches gluing the balls together. The construction of a triangular mesh of a smooth skin surface is often necessary for further analysis and for fast visualization. This package provides functions to construct a triangular mesh approximating the skin surface from a set of balls and a shrink factor. It also contains code to subdivide the mesh efficiently.
Introduced in: CGAL 3.3
Depends on: 3D Triangulations and 3D Polyhedral Surface
BibTeX: cgal:k-ssm3-16b

## 3D Mesh Generation

Pierre Alliez, Clément Jamin, Laurent Rineau, Stéphane Tayeb, Jane Tournois, Mariette Yvinec
This package is devoted to the generation of isotropic simplicial meshes discretizing 3D domains. The domain to be meshed is a region of 3D space that has to be bounded. The region may be connected or composed of multiple components and/or subdivided in several subdomains. The domain is input as an oracle able to answer queries, of a few different types, on the domain. Boundary and subdivision surfaces are either smooth or piecewise smooth surfaces, formed with planar or curved surface patches. Surfaces may exhibit 1-dimensional features (e.g. crease edges) and 0-dimensional features (e.g. singular points as corners tips, cusps or darts), that have to be fairly approximated in the mesh. Optionally, the algorithms support multi-core shared-memory architectures to take advantage of available parallelism.
Introduced in: CGAL 3.5
Depends on: 3D Triangulations
BibTeX: cgal:rty-m3-16b
Windows Demo: 3D Mesh Generation
Common Demo Dlls: dlls

# Shape Reconstruction

## Poisson Surface Reconstruction

Pierre Alliez, Laurent Saboret, Gaël Guennebaud
This package implements a surface reconstruction method: Poisson Surface Reconstruction. It takes as input a set of points with oriented normals and computes an implicit function. The CGAL surface mesh generator can then be used to extract an iso-surface from this function.
Introduced in: CGAL 3.5
Depends on: CGAL and Solvers
BibTeX: cgal:asg-srps-16b
Windows Demo: See Polyhedral Surface
Common Demo Dlls: dlls

## Scale-Space Surface Reconstruction

Thijs van Lankveld
This method allows to reconstruct a surface that interpolates a set of 3D points. This method provides an efficient alternative to the Poisson surface reconstruction method. The main difference in output is that this method reconstructs a surface that interpolates the point set (as opposed to approximating the point set). How the surface connects the points depends on a scale variable, which can be estimated semi-automatically.
Introduced in: CGAL 4.6
BibTeX: cgal:ssr3-16b
Depends on: 3D Alpha Shapes, dD Spatial Searching, CGAL and Solvers

Tran Kai Frank Da, David Cohen-Steiner
This package provides a greedy algorithm for surface reconstruction from an unorganized point set. Starting from a seed facet, a piecewise linear surface is grown by adding Delaunay triangles one by one. The most plausible triangles are added first, in a way that avoids the appearance of topological singularities.
Introduced in: CGAL 4.7
Depends on: 3D Triangulations
BibTeX: cgal:dc-afsr-16b

## Optimal Transportation Curve Reconstruction

Pierre Alliez, David Cohen-Steiner, Fernando de Goes, Clément Jamin, Ivo Vigan
This package provides an algorithm to reconstruct and simplify a shape from a point set in the plane, possibly hampered with noise and outliers. It generates as output a set of line segments and isolated points, which approximate the input point set.
Introduced in: CGAL 4.8
Depends on: 2D Triangulation
BibTeX: cgal:gavj-rs-16b

# Geometry Processing

## Polygon Mesh Processing

Sébastien Loriot, Jane Tournois, Ilker O. Yaz
This package provides a collection of methods and classes for polygon mesh processing, ranging from basic operations on simplices, to complex geometry processing algorithms.
Introduced in: CGAL 4.7
Depends on: documented for each function; CGAL and Solvers
BibTeX: cgal:lty-pmp-16b
Windows Demo: Operations on Polyhedra
Common Demo Dlls: dlls

## 3D Surface Subdivision Methods

Le-Jeng Andy Shiue
Subdivision methods recursively refine a control mesh and generate points approximating the limit surface. This package consists of four popular subdivision methods and their refinement hosts. Supported subdivision methods include Catmull-Clark, Loop, Doo-Sabin and sqrt(3) subdivisions. Their respective refinement hosts are Pqq, Ptq, Dqq and sqrt(3) refinements. Variations of those methods can be easily extended by substituting the geometry computation of the refinement host.
Introduced in: CGAL 3.2
BibTeX: cgal:s-ssm2-16b
Windows Demo: Operations on Polyhedra
Common Demo Dlls: dlls

## Triangulated Surface Mesh Segmentation

Ilker O. Yaz and Sébastien Loriot
This package provides a method to generate a segmentation of a triangulated surface mesh. The algorithm first computes the Shape Diameter Function (SDF) for all facets and applies a graph-cut based algorithm over these values. Low level functions are provided to replace any intermediate step by a custom one.
Introduced in: CGAL 4.4
Depends on: 3D Fast Intersection and Distance Computation
BibTeX: cgal:y-smsimpl-16b
Windows Demo: Operations on Polyhedra
Common Demo Dlls: dlls

## Triangulated Surface Mesh Simplification

Fernando Cacciola
This package provides an algorithm to simplify a triangulated surface mesh by edge collapsing. It is an implementation of the Turk/Lindstrom memoryless surface mesh simplification algorithm.
Introduced in: CGAL 3.3
BibTeX: cgal:c-tsms-12-16b
Windows Demo: Operations on Polyhedra
Common Demo Dlls: dlls

## Triangulated Surface Mesh Deformation

Sébastien Loriot, Olga Sorkine-Hornung, Yin Xu and Ilker O. Yaz
This package offers surface mesh deformation algorithms which provide new positions to the vertices of a surface mesh under positional constraints of some of its vertices, without requiring any additional structure other than the surface mesh itself.
Introduced in: CGAL 4.5
Depends on: CGAL and Solvers and Eigen
BibTeX: cgal:lsxy-tsmd-16b
Windows Demo: Edit plugin of the Polyhedron demo
Common Demo Dlls: dlls

## Triangulated Surface Mesh Parameterization

Laurent Saboret, Pierre Alliez and Bruno Lévy
Parameterizing a surface amounts to finding a one-to-one mapping from a suitable domain to the surface. In this package, we focus on triangulated surfaces that are homeomorphic to a disk and on piecewise linear mappings into a planar domain. This package implements several surface mesh parameterization methods, such as least squares conformal maps, discrete conformal map, discrete authalic parameterization, Floater mean value coordinates or Tutte barycentric mapping.
Introduced in: CGAL 3.2
Depends on: CGAL and Solvers
BibTeX: cgal:sal-pptsm2-16b
Windows Demo: Operations on Polyhedra
Common Demo Dlls: dlls

## Triangulated Surface Mesh Shortest Paths

Stephen Kiazyk, Sébastien Loriot, Éric Colin de Verdière
The package provides methods for computing geodesic shortest path on triangulated surface meshes. The algorithm used is based on a paper by Xin and Wang [11] . The input of this package can be any model of the FaceListGraph concept.
Introduced in: CGAL 4.7
BibTeX: cgal:klcdv-tsmsp-16b
Windows Demo: Operations on Polyhedra
Common Demo Dlls: dlls

## Triangulated Surface Mesh Skeletonization

Xiang Gao, Sébastien Loriot and Andrea Tagliasacchi
This package provides a (1D) curve skeleton extraction algorithm for a triangulated polygonal mesh without borders based on the mean curvature flow. The particularity of this skeleton is that it captures the topology of the input. For each skeleton vertex one can obtain its location and its corresponding vertices from the input mesh. The code is generic and works with any model of the FaceListGraph concept.
Introduced in: CGAL 4.7
Depends on: CGAL and Solvers
BibTeX: cgal:glt-tsms-16b
Windows Demo: See Polyhedral Surface
Common Demo Dlls: dlls

## Approximation of Ridges and Umbilics on Triangulated Surface Meshes

Marc Pouget and Frédéric Cazals
Global features related to curvature extrema encode important informations used in segmentation, registration, matching and surface analysis. Given pointwise estimations of local differential quantities, this package allows the approximation of differential features on a triangulated surface mesh. Such curvature related features are curves: ridges or crests, and points: umbilics.
Introduced in: CGAL 3.3
Depends on: CGAL and Solvers
BibTeX: cgal:cp-arutsm-16b

## Estimation of Local Differential Properties of Point-Sampled Surfaces

Marc Pouget and Frédéric Cazals
For a surface discretized as a point cloud or a mesh, it is desirable to estimate pointwise differential quantities. More precisely, first order properties correspond to the normal or the tangent plane; second order properties provide the principal curvatures and directions, third order properties provide the directional derivatives of the principal curvatures along the curvature lines, etc. This package allows the estimation of local differential quantities of a surface from a point sample.
Introduced in: CGAL 3.3
Depends on: CGAL and Solvers
BibTeX: cgal:pc-eldp-16b
Windows Demo: Operations on Polyhedra
Common Demo Dlls: dlls

## Point Set Processing

Pierre Alliez, Clément Jamin, Quentin Mérigot, Jocelyn Meyron, Laurent Saboret, Nader Salman, Shihao Wu
This CGAL component implements methods to analyze and process unorganized point sets. The input is an unorganized point set, possibly with normal attributes (unoriented or oriented). The point set can be analyzed to measure its average spacing, and processed through functions devoted to the simplification, outlier removal, smoothing, normal estimation, normal orientation and feature edges estimation.
Introduced in: CGAL 3.5
Depends on: CGAL and Solvers
BibTeX: cgal:ass-psp-16b
Windows Demo: See Polyhedral Surface
Common Demo Dlls: dlls

## Point Set Shape Detection

Sven Oesau, Yannick Verdie, Clément Jamin, Pierre Alliez
This component implements an efficient RANSAC-based primitive shape detection algorithm for point sets with unoriented normals. Five types of primitive shapes are detected: plane, sphere, cylinder, cone and torus. Other types of shapes can be detected through implementing a class deriving from the base shape class.
Introduced in: CGAL 4.7
BibTeX: cgal:ovja-pssd-16b
Windows Demo: Operations on Polyhedra
Common Demo Dlls: dlls

## 2D Placement of Streamlines

Abdelkrim Mebarki
Visualizing vector fields is important for many application domains. A good way to do it is to generate streamlines that describe the flow behavior. This package implements the "Farthest Point Seeding" algorithm for placing streamlines in 2D vector fields. It generates a list of streamlines corresponding to an input flow using a specified separating distance. The algorithm uses a Delaunay triangulation to model objects and address different queries, and relies on choosing the centers of the biggest empty circles to start the integration of the streamlines.
Introduced in: CGAL 3.2
Depends on: 2D Triangulation
BibTeX: cgal:m-ps-16b
Windows Demo: 2D Stream Lines
Common Demo Dlls: dlls

# Spatial Searching and Sorting

## 2D Range and Neighbor Search

Matthias Bäsken
This package supports circular, triangular, and isorectangular range search queries as well as (k) nearest neighbor search queries on 2D point sets. In contrast to the spatial searching package, this package uses a Delaunay triangulation as underlying data structure.
Introduced in: CGAL 2.1
Depends on: 2D Triangulation
BibTeX: cgal:b-ss2-16b

## Interval Skip List

Andreas Fabri
An interval skip list is a data structure for finding all intervals that contain a point, and for stabbing queries, that is for answering the question whether a given point is contained in an interval or not. For a triangulated terrain, this allows to quickly identify the triangles which intersect an iso line.
Introduced in: CGAL 3.0
BibTeX: cgal:f-isl-16b

## dD Spatial Searching

Hans Tangelder and Andreas Fabri
This package implements exact and approximate distance browsing by providing exact and approximate algorithms for range searching, k-nearest and k-furthest neighbor searching, as well as incremental nearest and incremental furthest neighbor searching, where the query items are points in dD Euclidean space.
Introduced in: CGAL 3.0
BibTeX: cgal:tf-ssd-16b
Windows Demo: 2D Spatial Searching
Common Demo Dlls: dlls

## dD Range and Segment Trees

Gabriele Neyer
Range and segment trees allow to perform window queries on point sets, and to enumerate all ranges enclosing a query point. The provided data structures are static and they are optimized for fast queries.
Introduced in: CGAL 0.9
BibTeX: cgal:n-rstd-16b

## Intersecting Sequences of dD Iso-oriented Boxes

Lutz Kettner, Andreas Meyer, and Afra Zomorodian
An efficient algorithm for finding all intersecting pairs for large numbers of iso-oriented boxes, in order to apply a user defined callback on them. Typically these boxes will be bounding boxes of more complicated geometries. The algorithm is useful for (self-) intersection tests of surfaces etc.
Introduced in: CGAL 3.1
BibTeX: cgal:kmz-isiobd-16b
Windows Demo: Operations on Polyhedra
Common Demo Dlls: dlls

## 3D Fast Intersection and Distance Computation

Pierre Alliez, Stéphane Tayeb, Camille Wormser
The AABB (axis-aligned bounding box) tree component offers a static data structure and algorithms to perform efficient intersection and distance queries on sets of finite 3D geometric objects.
Introduced in: CGAL 3.5
BibTeX: cgal:atw-aabb-16b
Windows Demo: AABB Tree
Common Demo Dlls: dlls

## Spatial Sorting

Christophe Delage and Olivier Devillers
This package provides functions for sorting geometric objects in two, three and higher dimensions, including on a sphere, in order to improve efficiency of incremental geometric algorithms.
Introduced in: CGAL 3.3
BibTeX: cgal:dd-ss-16b

# Geometric Optimization

## Bounding Volumes

Kaspar Fischer, Bernd Gärtner, Thomas Herrmann, Michael Hoffmann, and Sven Schönherr
This package provides algorithms for computing optimal bounding volumes of point sets. In d-dimensional space, the smallest enclosing sphere, ellipsoid (approximate), and annulus can be computed. In 3-dimensional space, the smallest enclosing strip is available as well, and in 2-dimensional space, there are algorithms for a number of additional volumes (rectangles, parallelograms, $$k=2,3,4$$ axis-aligned rectangles). The smallest enclosing sphere algorithm can also be applied to a set of d-dimensional spheres.
Introduced in: CGAL 1.1
BibTeX: cgal:fghhs-bv-16b
Windows Demo: 2D Bounding Volumes
Common Demo Dlls: dlls

## Inscribed Areas

Michael Hoffmann and Eli Packer
This package provides algorithms for computing inscribed areas. The algorithms for computing inscribed areas are: the largest inscribed k-gon (area or perimeter) of a convex point set and the largest inscribed iso-rectangle.
Introduced in: CGAL 1.1
BibTeX: cgal:hp-ia-16b
Windows Demos: 2D Inscribed k-gon as part of Polygon demo, 2D Largest Empty Rectangle
Common Demo Dlls: dlls

## Optimal Distances

Kaspar Fischer, Bernd Gärtner, Thomas Herrmann, Michael Hoffmann, and Sven Schönherr
This package provides algorithms for computing the distance between the convex hulls of two point sets in d-dimensional space, without explicitely constructing the convex hulls. It further provides an algorithm to compute the width of a point set, and the furthest point for each vertex of a convex polygon.
Introduced in: CGAL 1.1
BibTeX: cgal:fghhs-od-16b

## Principal Component Analysis

Pierre Alliez, Sylvain Pion and Ankit Gupta
This package provides functions to compute global information about the shape of a set of 2D or 3D objects. It provides the computation of axis-aligned bounding boxes for point sets, and barycenters of weighted point sets. In addition, it provides computation of centroids (center of mass) and linear least squares fitting for point sets as well as for sets of other bounded objects. More specifically, it is possible to fit 2D lines to 2D segments, circles, disks, iso rectangles and triangles, as well as to fit 3D lines or 3D planes to 3D segments, triangles, iso cuboids, tetrahedra, spheres and balls. The common interface to these functions takes an iterator range of objects.
Introduced in: CGAL 3.2
Depends on: CGAL and Solvers
Windows Demos: Principal Component Analysis, Operations on Polygons, Operations on Polyhedra
Common Demo Dlls: dlls

# Interpolation

## 2D and Surface Function Interpolation

Julia Flötotto
This package implements different methods for scattered data interpolation: Given measures of a function on a set of discrete data points, the task is to interpolate this function on an arbitrary query point. The package further offers functions for natural neighbor interpolation.
Introduced in: CGAL 3.1
BibTeX: cgal:f-i-16b

## 2D Generalized Barycentric Coordinates

Dmitry Anisimov, David Bommes, Kai Hormann, and Pierre Alliez
The package 2D Generalized Barycentric Coordinates offers an efficient and robust implementation of two-dimensional closed-form generalized barycentric coordinates defined for simple two-dimensional polygons. If coordinates with respect to multivariate scattered points instead of a polygon are required, please refer to natural neighbor coordinates from the Package 2D and Surface Function Interpolation.
Introduced in: CGAL 4.6
BibTeX: cgal:abha-gbc-16b

# Kinetic Data Structures

## Kinetic Data Structures

Daniel Russel
Kinetic data structures allow combinatorial structures to be maintained as the primitives move. The package provides implementations of kinetic data structures for Delaunay triangulations in two and three dimensions, sorting of points in one dimension and regular triangulations in three dimensions. The package supports exact or inexact operations on primitives which move along polynomial trajectories.
Introduced in: CGAL 3.2
BibTeX: cgal:r-kds-16b

## Kinetic Framework

Daniel Russel
Kinetic data structures allow combinatorial geometric structures to be maintained as the primitives move. The package provides a framework to ease implementing and debugging kinetic data structures. The package supports exact or inexact operations on primitives which move along polynomial trajectories.
Introduced in: CGAL 3.2
BibTeX: cgal:r-kdsf-16b

# Support Library

## STL Extensions for CGAL

Michael Hoffmann, Lutz Kettner, Sylvain Pion, and Ron Wein
CGAL is designed in the spirit of the generic programming paradigm to work together with the Standard Template Library (STL). This package provides non-geometric STL-like algorithms and datastructures that are not in the STL standard, as well as functions to change the failure behaviour of assertions.
Introduced in: CGAL 1.0
BibTeX: cgal:hkpw-se-16b

## CGAL and the Boost Graph Library

Andreas Fabri, Fernando Cacciola, Philipp Moeller, and Ron Wein
This package provides a framework for interfacing CGAL data structures with the algorithms of the Boost Graph Library, or BGL for short. It allows to run graph algorithms directly on CGAL data structures which are model of the BGL graph concepts, for example the shortest path algorithm on a Delaunay triangulation in order to compute the Euclidean minimum spanning tree. Furthermore, it introduces several new graph concepts describing halfedge data structures.
Introduced in: CGAL 3.3
BibTeX: cgal:cfw-cbgl-16b

## CGAL and Solvers

Simon Giraudot, Pierre Alliez, Frédéric Cazals, Gaël Guennebaud, Bruno Lévy, Marc Pouget, and Laurent Saboret
This package provides concepts and models for solving linear systems with dense or sparse matrices.
Introduced in: CGAL 4.8
BibTeX: cgal:eb-solver-16b

## CGAL and Boost Property Maps

Andreas Fabri and Laurent Saboret
This package provides a framework for interfacing CGAL data structures with algorithms expecting Boost Property Maps.
Introduced in: CGAL 3.5
BibTeX: cgal:fs-cbpm-16b

## Cone-Based Spanners

Weisheng Si and Quincy Tse
This package provides functors for constructing two kinds of cone-based spanners: Yao graph and Theta graph, given a set of vertices on the plane and the directions of cone boundaries. Both exact and inexact constructions are supported. In exact construction, the cone boundaries are calculated using the roots of polynomials, thus avoiding the use of $$\pi$$, which cannot be represented exactly. In inexact construction, the cone boundaries are calculated using the approximate $$\pi$$ value defined in CGAL, which is still accurate enough for most applications. Moreover, for visualization purpose, this package provides a global function to generate the data and script files used by Gnuplot to plot the constructed graphs.
Introduced in: CGAL 4.9
BibTeX: cgal:st-cbs-16b

## Handles and Circulators

Olivier Devillers, Lutz Kettner, Sylvain Pion, Michael Seel, and Mariette Yvinec
This package descibes handles and circulators. They are related to iterators. Handles allow to dereference but neither to increment nor to decrement. Circulators have no notion of past-the-end, and they are used in CGAL whenever we have cyclic stuctures.
Introduced in: CGAL 1.0
BibTeX: cgal:dksy-hc-16b

## Geometric Object Generators

Pedro M. M. de Castro, Olivier Devillers, Susan Hert, Michael Hoffmann, Lutz Kettner, Sven Schönherr, and Alexandru Tifrea
This package provides a variety of generators for geometric objects. They are useful as synthetic test data sets, e.g. for testing algorithms on degenerate object sets and for performance analysis.
Introduced in: CGAL 1.0
BibTeX: cgal:dhhk-gog-16b
Windows Demo: Generators
Common Demo Dlls: dlls

## Profiling tools, Hash Map, Union-find, Modifiers

Lutz Kettner, Sylvain Pion, and Michael Seel
This package provides classes for profiling time and memory consumption, profiling macros, a hash map, a union find data structure and a modifier.
Introduced in: CGAL 3.2
BibTeX: cgal:kps-pthum-16b

## IO Streams

Andreas Fabri, Geert-Jan Giezeman, and Lutz Kettner
All classes in the CGAL kernel provide input and output operators for IO streams. The basic task of such an operator is to produce a representation of an object that can be written as a sequence of characters on devices as a console, a file, or a pipe. In CGAL we distinguish between a raw ascii, a raw binary and a pretty printing format.
Introduced in: CGAL 1.0
BibTeX: cgal:fgk-ios-12-16b

# Visualization

## Geomview

Andreas Fabri and Sylvain Pion
This package implements an interface to Geomview, an interactive 3D viewing program, originally developed at the Geometry Center in Minneapolis.
Introduced in: CGAL 2.0
BibTeX: cgal:fp-gv-16b

## CGAL and the Qt Graphics View Framework

Andreas Fabri and Laurent Rineau
This package provides classes for displaying CGAL objects and data structures in the Qt 5 Graphics View Framework.
Introduced in: CGAL 3.4
Depends on: Qt 5
BibTeX: cgal:fr-cqgvf-16b